Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{p - 3}{p + 1} \times \dfrac{p^2 - 2p - 3}{p^2 - 3p} $
First factor the quadratic. $r = \dfrac{p - 3}{p + 1} \times \dfrac{(p + 1)(p - 3)}{p^2 - 3p} $ Then factor out any other terms. $r = \dfrac{p - 3}{p + 1} \times \dfrac{(p + 1)(p - 3)}{p(p - 3)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ (p - 3) \times (p + 1)(p - 3) } { (p + 1) \times p(p - 3) } $ $r = \dfrac{ (p - 3)(p + 1)(p - 3)}{ p(p + 1)(p - 3)} $ Notice that $(p - 3)$ and $(p + 1)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ (p - 3)\cancel{(p + 1)}(p - 3)}{ p\cancel{(p + 1)}(p - 3)} $ We are dividing by $p + 1$ , so $p + 1 \neq 0$ Therefore, $p \neq -1$ $r = \dfrac{ (p - 3)\cancel{(p + 1)}\cancel{(p - 3)}}{ p\cancel{(p + 1)}\cancel{(p - 3)}} $ We are dividing by $p - 3$ , so $p - 3 \neq 0$ Therefore, $p \neq 3$ $r = \dfrac{p - 3}{p} ; \space p \neq -1 ; \space p \neq 3 $